A383776 a(n) = (11*n + 3 + 6/(n+2)) * Catalan(n).

6, 16, 53, 186, 672, 2472, 9207, 34606, 130988, 498576, 1906346, 7316596, 28170768, 108760560, 420889995, 1632155670, 6340808820, 24673450560, 96148670310, 375164728620, 1465589068320, 5731488987120, 22436098732710, 87905595401676, 344702077523352, 1352701532137312, 5312100899224532, 20874451526714856

Offset: 0

Author

F. Chapoton, May 09 2025

Keywords

• nonn
• easy

Comments

It appears that for n >= 2 a(n-2) is the number of lattice points in the n-dimensional lattice polytope defined, in the space with coordinates (x_1,x_2,...,x_n), by the equations x_i >= 0 for every i, sum_i x_i <= n and x_1 + x_2 <= 2. For n=2, this is a triangle with 6 lattice points.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A000984, A007054, A051960.

Programs

Mathematica

A383776[n_] := (11*n + 3 + 6/(n + 2))*CatalanNumber[n];
Array[A383776, 30, 0] (* Paolo Xausa, May 15 2025 *)

Sage

[(11*n+3+6/(n+2))*catalan_number(n) for n in range(12)]

Formula

a(n) = (11*n + 3 + 6/(n + 2))*Catalan(n).

G.f.: 2*(7 + 5*sqrt(1 - 4*x) - 6*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)). - Stefano Spezia, May 15 2025